The binary actions of simple groups of Lie type of characteristic 2

Nick Gill, Pierre Guillot and Martin W. Liebeck

Vol. 336 (2025), No. 1-2, 113–135

DOI: 10.2140/pjm.2025.336.113

Abstract

Let $\mathcal{𝒞}$ be a conjugacy class of involutions in a group $G$. We study the graph $\Gamma (\mathcal{𝒞})$ whose vertices are elements of $\mathcal{𝒞}$ with $g,h\in \mathcal{𝒞}$ connected by an edge if and only if $gh\in \mathcal{𝒞}$. For $t\in \mathcal{𝒞}$, we define the component group of $t$ to be the subgroup of $G$ generated by all vertices in $\Gamma (\mathcal{𝒞})$ that lie in the connected component of the graph that contains $t$.

We classify the component groups of all involutions in simple groups of Lie type over a field of characteristic $2$. We use this classification to partially classify the transitive binary actions of the simple groups of Lie type over a field of characteristic $2$ for which a point stabiliser has even order. The classification is complete unless the simple group in question is a symplectic or unitary group.

In memory of Gary Seitz

Keywords

permutation group, relational complexity, binary action, group of Lie type

Mathematical Subject Classification

Primary: 20D06

Secondary: 20E45

Milestones

Received: 28 March 2024

Revised: 13 November 2024

Accepted: 14 November 2024

Published: 26 May 2025